I just got off the phone with the CT DESPP, the lady I spoke to confirmed that the Social Security number is optional and leaving it blank is okay. I had other questions about the form; - she suggested I contact the legislature about them, her implication was that the form and questions were created by politicians and not the DESPP. I went to review the act, it seems that most of the questions including the social security one were generated by the DESPP, so they are mis-characterizing their own product. Here is the relevant part in the act
"Sec. 28. Section 53-202d of the general statutes is repealed and the following is substituted in lieu thereof (Effective from passage)4) The certificate of possession shall contain a description of the firearm that identifies it uniquely, including all identification marks, the full name, address, date of birth and thumbprint of the owner, and any other information as the department may deem appropriate.
As you can see the act asks for little (relatively), the DESPP have added social security, home telephone number, sex, height, weight, operator license number and importer.